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How many factors of the number 2^3 x 3 x 5^2 are perfect squares?

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How many factors of the number 2^3 x 3 x 5^2 are perfect squares?  [#permalink]

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New post Updated on: 13 Aug 2018, 02:08
1
5
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A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

32% (01:00) correct 68% (00:57) wrong based on 185 sessions

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:40.
Last edited by EgmatQuantExpert on 13 Aug 2018, 02:08, edited 4 times in total.
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Re: How many factors of the number 2^3 x 3 x 5^2 are perfect squares?  [#permalink]

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New post 27 Feb 2018, 09:56
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The answer should be 4 as per the logic explained in the below link by various moderators:

https://gmatclub.com/forum/how-many-factors-of-10800-are-perfect-squares-251823.html
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Re: How many factors of the number 2^3 x 3 x 5^2 are perfect squares?  [#permalink]

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New post 27 Feb 2018, 22:36
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Solution


    • Per our conceptual knowledge, a number is a perfect square only when the exponents of the prime factors in the number are even.
      o For example: \(7^2\)is a perfect square but \(5^3\) is not, since the exponent of \(7\)is even but the exponent of \(5\)is not.
    • To find the factors which are perfect squares we need to first of focus on each prime number and its powers and try to find out, what are the possible perfect squares that we can create using those prime numbers.
    • So, for making perfect squares, we need even powers of the prime factors and this can be written in the form: \(2^a * 3^b * 5^c\) [where a, b and c can be 0,2,4,8…]
      o The factors of\(2^3\) are: \(1, 2, 4, 8\) out of which only \(1(2^0)\) and \(4(2^2)\) are perfect squares.
      o The factors of \(3\) are: \(1\) and \(3\), out of which only \(1(3^0)\) is a perfect square.
      o The factors of \(5^2\)are: \(1, 5, 25,\) out of which only \(1(5^0)\) and \(25(5^2)\) are perfect squares.
    • Thus, the possible unique perfect square factors using only these prime numbers are \(1\), \(4\) and \(25\).
    But is that all?
    Look at the following factor:
    • Is \(2^2\) *\(5^2\) a perfect square?
      o Yes, it is.
    So, the criterion of perfect squares also holds true when we multiply 2 or more such perfect squares together.
      o Hence, 100 will also be included in our list.
So, there are 4 possible cases where the factors of the given number are perfect squares.
Hence, the correct answer is Option D.
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Re: How many factors of the number 2^3 x 3 x 5^2 are perfect squares?  [#permalink]

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New post 06 May 2019, 09:42
Any factor of 2^3∗3∗5^2 can be written as 2^a* 3^b* 5^c
For factor to be a perfect square-
a can have 2 values (0,2)
b can have 1 value (0)
c can have 2 values (0,2)
Total factors of the number 2^3∗3∗5^2 are perfect squares= 2*1*2=4
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Re: How many factors of the number 2^3 x 3 x 5^2 are perfect squares?   [#permalink] 06 May 2019, 09:42
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